In Section 3, we devise a theory of ensembles for which the corresponding intensity measurements are “almost” injective, that is, \(\mathcal {A}^{-1}(\mathcal {A}(x))=\lbrace \omega x:|\omega |=1\rbrace \)  for almost every \(x\) . In this section, we focus on the real case, meaning phase retrieval is up to a global sign factor \(\omega =\pm 1\) , and our approach is inspired by the characterization of injectivity in the real case by Balan, Casazza and Edidin [1]}.
After characterizing almost injectivity in the real case, we find a particularly satisfying sufficient condition for almost injectivity: that \(\Phi \)  forms a unit norm tight frame with \(M\)  and \(N\)  relatively prime.
Characterizing almost injectivity in the complex case remains an open problem.
